What do echo state networks give us over a generic RNN resevoir?
The echo state condition states that differences in the input sequence results in separate trajectories of reservoir states.
Which means that when you make the reservoir large enough any difference in the input sequence result in a linearly separable difference in the state space.
This is quite similar to the Kerneltrick in e.g. Support Vector machines where the data becomes linearly separable in the feature space. In reservoir computing the reservoir is in a way a random feature space.
To sum up, the echo state condition ensures that the signal becomes linearly separable.